The plectic conjecture over local fields

Abstract:

The étale cohomology of varieties over Q enjoys a Galois action. For Hilbert modular varieties, Nekovář-Scholl observed that this Galois action on the level of cohomology extends to a much larger profinite group: the plectic group. Motivated by applications to higher-rank Euler systems, they conjectured that this extension holds even on the level of complexes, as well as for more general Shimura varieties.

We present a proof of the analog of this conjecture for local Shimura varieties. Consequently, we obtain results for the basic locus of global Shimura varieties, after restricting to a decomposition group. The proof crucially uses a mixed-characteristic version of fusion due to Fargues–Scholze.